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manuscripta mathematica

, Volume 151, Issue 3–4, pp 375–418 | Cite as

Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients

  • Ariel BartonEmail author
Article

Abstract

This paper considers the theory of higher-order divergence-form elliptic differential equations. In particular, we provide new generalizations of several well-known tools from the theory of second-order equations. These tools are the Caccioppoli inequality, Meyers’s reverse Hölder inequality for gradients, and the fundamental solution. Our construction of the fundamental solution may also be of interest in the theory of second-order operators, as we impose no regularity assumptions on our elliptic operator beyond ellipticity and boundedness of coefficients.

Mathematics Subject Classification

35J48 31B10 35C15 

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References

  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17, 35–92 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agranovich, M.S.: On the theory of Dirichlet and Neumann problems for linear strongly elliptic systems with Lipschitz domains. Funktsional. Anal. i Prilozhen. 41(4), 1–21, 96 (2007). doi: 10.1007/s10688-007-0023-x. English translation: Funct. Anal. Appl. 41(4), 247–263 (2007)
  3. 3.
    Auscher, P., Hofmann, S., McIntosh, A., Tchamitchian, P.: The Kato square root problem for higher order elliptic operators and systems on \({\mathbb{R}}^n\). J. Evol. Equ. 1(4), 361–385 (2001). doi: 10.1007/PL00001377. Dedicated to the memory of Tosio KatoMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Auscher, P., McIntosh, A., Tchamitchian, P.: Heat kernels of second order complex elliptic operators and applications. J. Funct. Anal. 152(1), 22–73 (1998). doi: 10.1006/jfan.1997.3156 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Auscher, P., Qafsaoui, M.: Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form. J. Funct. Anal. 177(2), 310–364 (2000). doi: 10.1006/jfan.2000.3643 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Babuška, I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1970/1971)Google Scholar
  7. 7.
    Barton, A., Hofmann, S., Mayboroda, S.: Square function estimates on layer potentials for higher-order elliptic equations. ArXiv e-prints (2015). arXiv:1508.04988 [math.AP]
  8. 8.
    Barton, A., Mayboroda, S.: Higher-order elliptic equations in non-smooth domains: a partial survey. In: Harmonic analysis, partial differential equations, complex analysis, banach spaces, andoperator theory. Celebrating Cora Sadosky’s life, vol. 1. AWM-Springer (2016) (To appear)Google Scholar
  9. 9.
    Campanato, S.: Sistemi ellittici in forma divergenza. Regolarità all’interno. Quaderni. [Publications]. Scuola Normale Superiore Pisa, Pisa (1980)Google Scholar
  10. 10.
    Cho, S., Dong, H., Kim, S.: Global estimates for Green’s matrix of second order parabolic systems with application to elliptic systems in two dimensional domains. Potential Anal. 36(2), 339–372 (2012). doi: 10.1007/s11118-011-9234-0 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cohen, J., Gosselin, J.: Adjoint boundary value problems for the biharmonic equation on \(C^1\) domains in the plane. Ark. Mat. 23(2), 217–240 (1985). doi: 10.1007/BF02384427 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dalla Riva, M.: A family of fundamental solutions of elliptic partial differential operators with real constant coefficients. Integral Equ. Oper. Theory 76(1), 1–23 (2013). doi: 10.1007/s00020-013-2052-6 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dalla Riva, M., Morais, J., Musolino, P.: A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients. Math. Methods Appl. Sci. 36(12), 1569–1582 (2013). doi: 10.1002/mma.2706 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dolzmann, G., Müller, S.: Estimates for Green’s matrices of elliptic systems by \(L^p\) theory. Manuscr. Math. 88(2), 261–273 (1995). doi: 10.1007/BF02567822 CrossRefzbMATHGoogle Scholar
  15. 15.
    Dong, H., Kim, S.: Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains. Trans. Am. Math. Soc. 361(6), 3303–3323 (2009). doi: 10.1090/S0002-9947-09-04805-3 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Duduchava, R.: The Green formula and layer potentials. Integral Equ. Oper. Theory 41(2), 127–178 (2001). doi: 10.1007/BF01295303 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)Google Scholar
  18. 18.
    Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Frehse, J.: An irregular complex valued solution to a scalar uniformly elliptic equation. Calc. Var. Partial Differ. Equ. 33(3), 263–266 (2008). doi: 10.1007/s00526-007-0131-8 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Friedman, A.: On fundamental solutions of elliptic equations. Proc. Am. Math. Soc. 12, 533–537 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fuchs, M.: The Green matrix for strongly elliptic systems of second order with continuous coefficients. Z. Anal. Anwendungen 5(6), 507–531 (1986)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton, NJ (1983)zbMATHGoogle Scholar
  23. 23.
    Grüter, M., Widman, K.O.: The Green function for uniformly elliptic equations. Manuscr. Math. 37(3), 303–342 (1982). doi: 10.1007/BF01166225 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hofmann, S., Kim, S.: The Green function estimates for strongly elliptic systems of second order. Manuscr. Math. 124(2), 139–172 (2007). doi: 10.1007/s00229-007-0107-1 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    John, F.: Plane waves and spherical means applied to partial differential equations. Interscience Publishers, New York-London (1955)zbMATHGoogle Scholar
  26. 26.
    Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1–2), 71–88 (1981). doi: 10.1007/BF02392869 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kang, K., Kim, S.: Global pointwise estimates for Green’s matrix of second order elliptic systems. J. Differ. Equ. 249(11), 2643–2662 (2010). doi: 10.1016/j.jde.2010.05.017 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kenig, C.E., Ni, W.M.: On the elliptic equation \(Lu-k+K\,{\rm exp}[2u]=0\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(2), 191–224 (1985). http://www.numdam.org/item?id=ASNSP_1985_4_12_2_191_0
  29. 29.
    Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 3(17), 43–77 (1963)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Mayboroda, S., Maz’ya, V.: Boundedness of the Hessian of a biharmonic function in a convex domain. Commun. Partial Differ. Equ. 33(7–9), 1439–1454 (2008). doi: 10.1080/03605300801891919 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mayboroda, S., Maz’ya, V.: Pointwise estimates for the polyharmonic Green function in general domains. In: Analysis, partial differential equations and applications, Oper. Theory Adv. Appl., vol. 193, pp. 143–158. Birkhäuser Verlag, Basel (2009)Google Scholar
  32. 32.
    Maz’ya, V.: The Wiener test for higher order elliptic equations. Duke Math. J. 115(3), 479–512 (2002). doi: 10.1215/S0012-7094-02-11533-6 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Maz’ya, V., Mitrea, M., Shaposhnikova, T.: The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients. J. Anal. Math. 110, 167–239 (2010). doi: 10.1007/s11854-010-0005-4 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Meyers, N.G.: An \(L^{p}\)-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 3(17), 189–206 (1963)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mitrea, I., Mitrea, M.: Multi-layer potentials and boundary problems for higher-order elliptic systems in Lipschitz domains. Lecture Notes in Mathematics, Vol. 2063. Springer, Heidelberg (2013)Google Scholar
  36. 36.
    Morrey Jr., C.B.: Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer, New York Inc., New York (1966)Google Scholar
  37. 37.
    Moser, J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)CrossRefzbMATHGoogle Scholar
  38. 38.
    Ortner, N., Wagner, P.: A survey on explicit representation formulae for fundamental solutions of linear partial differential operators. Acta Appl. Math. 47(1), 101–124 (1997). doi: 10.1023/A:1005784017770 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pipher, J., Verchota, G.C.: Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators. Ann. Math. 142(1), 1–38 (1995). doi: 10.2307/2118610 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Rosén, A.: Layer potentials beyond singular integral operators. Publ. Mat. 57(2), 429–454 (2013). doi: 10.5565/PUBLMAT_57213_08 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Verchota, G.C.: Potentials for the Dirichlet problem in Lipschitz domains. In: Potential theory—ICPT 94 (Kouty, 1994), pp. 167–187. de Gruyter, Berlin (1996)Google Scholar
  42. 42.
    Verchota, G.C.: The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194(2), 217–279 (2005). doi: 10.1007/BF02393222 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Verchota, G.C.: Boundary coerciveness and the Neumann problem for 4th order linear partial differential operators. In: Around the research of Vladimir Maz’ya. II, Int. Math. Ser. (N.Y.), Vol. 12, pp. 365–378. Springer, New York (2010). doi: 10.1007/978-1-4419-1343-2_17

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematical Sciences309 SCEN, University of ArkansasFayettevilleUSA

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